MR diffusion kurtosis imaging for neural tissue characterization y

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1 Special Issue Review Article Received: 15 September 2009, Revised: 10 December 2009, Accepted: 29 December 2009, Published online in Wiley Online Library: 9 July 2010 (wileyonlinelibrary.com) DOI: /nbm.1506 MR diffusion kurtosis imaging for neural tissue characterization y Ed X. Wu a,b * and Matthew M. Cheung a,b In conventional diffusion tensor imaging (DTI), water diffusion distribution is described as a 2nd-order threedimensional (3D) diffusivity tensor. It assumes that diffusion occurs in a free and unrestricted environment with a Gaussian distribution of diffusion displacement, and consequently that diffusion weighted (DW) signal decays with diffusion factor (b-value) monoexponentially. In biological tissue, complex cellular microstructures make water diffusion a highly hindered or restricted process. Non-monoexponential decays are experimentally observed in both white matter and gray matter. As a result, DTI quantitation is b-value dependent and DTI fails to fully utilize the diffusion measurements that are inherent to tissue microstructure. Diffusion kurtosis imaging (DKI) characterizes restricted diffusion and can be readily implemented on most clinical scanners. It provides a higher-order description of water diffusion process by a 2nd-order 3D diffusivity tensor as in conventional DTI together with a 4th-order 3D kurtosis tensor. Because kurtosis is a measure of the deviation of the diffusion displacement profile from a Gaussian distribution, DKI analyses quantify the degree of diffusion restriction or tissue complexity without any biophysical assumption. In this work, the theory of diffusion kurtosis and DKI including the directional kurtosis analysis is revisited. Several recent rodent DKI studies from our group are summarized, and DKI and DTI compared for their efficacy in detecting neural tissue alterations. They demonstrate that DKI offers a more comprehensive approach than DTI in describing the complex water diffusion process in vivo. By estimating both diffusivity and kurtosis, it may provide improved sensitivity and specificity in MR diffusion characterization of neural tissues. Copyright ß 2010 John Wiley & Sons, Ltd. Keywords: MRI; DTI; diffusion weighted signal; diffusion tensor imaging; DKI; diffusion kurtosis imaging; kurtosis; restricted diffusion; neural tissue; tissue characterization 836 INTRODUCTION Magnetic resonance (MR) diffusion tensor imaging (DTI) has been shown to provide unique microstructural information in characterizing tissue microanatomy (1 3) that cannot be offered non-invasively by other modalities. DTI describes the water diffusion process in tissue by a 2nd-order three-dimensional (3D) diffusivity tensor (DT) where the three diffusivity eigenvectors correspond to the axes of a tri-axial diffusivity ellipsoid (2). Typical DTI indices, derived from DT as rotationally invariant parameters, are mean diffusivity (MD), fractional anisotropy (FA), axial diffusivity (l // ) and radial diffusivity (l? ) (3). As water diffusion is anisotropic in nerve fibers due to myelination and other inherent axonal structures (4), DTI has demonstrated remarkable success in characterizing white matter integrity in normal and pathological states, and in describing orientational neuroarchitecture and connectivity in the central nervous system (CNS). Despite these accomplishments, conventional DTI fails to fully utilize the MR diffusion measurements that are inherent to tissue microstructure. DTI computes apparent diffusivity based on the assumption that diffusion weighted (DW) MR signal has a monoexponential dependence on the diffusion factor (b-value). DTI implicitly assumes that water molecule diffusion occurs in a free and unrestricted environment with a Gaussian distribution of diffusion displacement. This assumption has been experimentally demonstrated to be invalid in both white matter (WM) and gray matter (GM) when high b-values are used (5,6). In biological tissue, complex underlying cellular components and structures hinder and restrict the diffusion of water molecules. Such restricted or non-gaussian diffusion leads to the fact that DTI NMR Biomed. 2010; 23: * Correspondence to: E. X. Wu, Laboratory of Biomedical Imaging and Signal Processing, Departments of Electrical and Electronic Engineering, Medicine and Anatomy, The University of Hong Kong, Pokfulam, Hong Kong SAR, China. ewu@eee.hku.hk a b y E. X. Wu, M. M. Cheung Laboratory of Biomedical Imaging and Signal Processing, The University of Hong Kong, Pokfulam, Hong Kong SAR, China E. X. Wu, M. M. Cheung Department of Electrical and Electronic Engineering, The University of Hong Kong, Pokfulam, Hong Kong SAR, China This article is published in NMR in Biomedicine as a special issue on Progress in Diffusion-Weighted Imaging: Concepts, Techniques, and Applications to the Central Nervous System, edited by Jens H. Jensen and Joseph A. Helpern, Center for Biomedical Imaging, Department of Radiology, NYU School of Medicine, New York, NY, USA. Abbreviations used: D, diffusion time; l //, axial diffusivity; l 1, Largest eigenvalue of DT; l 2, intermediate eigenvalue of DT; l 3, smallest eigenvalue of DT; l?, radial diffusivity; k i, cumulants; d, duration of gradient pulse; m i, central moments; g, gyromagnetic ratio; 3D, three-dimensional; AC, anterior commissure; b-value, diffusion factor; CC, corpus callosum; CNS, central nervous system; CP, cerebral peduncle; CPu, caudate putamen; CSF, cerebrospinal fluid; CT, cerebral cortex; D, diffusion coefficient; D app, apparent diffusivity; DEC-FA, directionally-encoded colour FA; DKI, diffusion kurtosis imaging; DT, diffusivity tensor; DTI, diffusion tensor imaging; DW, diffusionweighted; EAE, experimental autoimmune encephalitis; EC, external capsule; EPI, echo-planar imaging; FA, fractional anisotropy; FA K, fractional anisotropy of kurtosis; FOV, field of view; g, strength of gradient pulse; GDTI, generalized DTI; GM, gray matter; HP, hippocampus; IC, internal capsule; K, excess kurtosis; K //, axial kurtosis; K?, radial kurtosis; K app, apparent kurtosis; KT, kurtosis tensor; MD, mean diffusivity; MK, mean kurtosis; MR, magnetic resonance; MS, multiple sclerosis; NEX, number of signal averages; P, displacement probability; PGSE, pulsed gradient spin echo; ROIs, regions of interest; S, signal intensity; SC, spinal cord; SE, spin echo; SNR, signal to noise ratio; TE, echo time; TR, repetition time; T2W, T2-weighted; VBM, voxel based morphometry; WM, white matter. Copyright ß 2010 John Wiley & Sons, Ltd.

2 DIFFUSION KURTOSIS IMAGING OF NEURAL TISSUE quantitation is b-value dependent, complicating quantitative and comparative studies. Water diffusion restriction in vivo is a complex process with numerous potential determinants such as cell structures, restricted intra- or extra-cellular compartments, permeability or water exchange, and potentially other biophysical properties associated with different water molecule populations. Moreover, the simplified description of the diffusion process in vivo by a 2nd-order 3D diffusivity tensor prevents DTI from being truly effective in characterizing relatively isotropic tissue such as GM. Even in WM, the DTI model can fail if the tissue contains substantial crossing or diverging fibers (7). Several approaches have been proposed to investigate this nonmonoexponential b-value dependence. Because there are at least two types of compartments (intra-cellular and extra-cellular) in tissue, a bi-exponential model was proposed (5,8,9). Despite good fits of DW signal attenuation, the estimated volume fractions of the fast and slow diffusion components are found to be inconsistent with the known ratio between the intra-cellular and extra-cellular compartments (5,8 11). More general approaches have been proposed by including higher order diffusion terms. Q-space imaging estimates the water diffusion displacement probability profile (12) and has shown that microstructural changes in diseased neural tissues can be detected (13,14). Although the q-space approach can fully describe the diffusion profile, it requires a prohibitively long scan time and is demanding with respect to gradient hardware. Stretched exponential model has also been developed to characterize the signal decay (15 17). Generalized DTI (GDTI) has also been formulated by adding a series of high-order diffusion tensors in the Bloch-Torrey diffusion equations (18,19). Diffusion kurtosis imaging (DKI) has been developed recently to probe non-gaussian diffusion properties (20 22). DW signal decay is related to the b-value in DKI by both the apparent diffusivity (D app ) and apparent kurtosis (K app ) in a quadratic exponential manner. Similar to determining the higher orders in GDTI (23), DKI is a truncation of the logarithmic expansion of the DW signal decay. In theory, DKI provides a higher-order description of restricted water diffusion process by a 2nd-order 3D diffusivity tensor (DT as in conventional DTI) together with a 4th-order 3D kurtosis tensor (KT). Kurtosis here refers to the excess kurtosis that is the normalized and standardized fourth central moment of the water displacement distribution (24). It is a dimensionless measure that quantifies the deviation of the water diffusion displacement profile from the Gaussian distribution of unrestricted diffusion, providing a measure of the degree of diffusion hindrance or restriction. Note that kurtosis is one of the parameters that can be obtained from q-space imaging (25 27), and it has been shown to offer superior sensitivity over conventional DTI (28). Recent DKI findings are promising. Mean kurtosis (MK), the average apparent kurtosis along all diffusion gradient encoding directions, has been measured and demonstrated to offer an improved sensitivity in detecting developmental and pathological changes in neural tissues as compared to conventional DTI (22,29 36). In addition, directional kurtosis analysis has been formulated to reveal directionally specific information, such as the water diffusion kurtoses along the direction parallel or perpendicular to the principle water diffusion direction as determined by the 2nd-order diffusion tensor (22,33). In this review, we will first revisit the theory of kurtosis, DKI and directional kurtosis analysis. We will summarize several rodent DKI studies from our group, and compare the efficacy of DKI and DTI for inferring microstructural information in neural tissues. Finally, we will discuss the issues related to DKI quantitation and interpretation. THEORY DIFFUSION KURTOSIS IMAGING AND DIRECTIONAL ANALYSIS Diffusion kurtosis Mathematically, excess kurtosis (K) is defined by (37): K ¼ m 4 m 2 3 ¼ k 4 2 k 2 (1) 2 where k i and m i are the i th -order cumulant and central moment of the distribution. Cumulants can be expressed by central moments. In particular, the first three cumulants have same expressions as the central moments, i.e., k 1, k 2 and k 3 are equal to m, m 2 and m 3, respectively. m and m 2 are the mean and variance of the distribution, and m 3 is the skewness. K is related to k 4 as stated in Eq. (1). K is a dimensionless metric that measures the deviation of the water diffusion profile from a Gaussian function for which K ¼ 0. Figure 1 illustrates the Gaussian function and two related functions. A positive K means that the distribution is more sharply peaked than a Gaussian, such as Gaussian squared. In conventional DTI, the diffusion displacement distribution of water molecules is assumed to be Gaussian and the diffusion coefficient can be estimated from a pulsed gradient spin echo (PGSE) sequence as (38,39): ln SðbÞ ¼ bd (2) Sð0Þ where b is the diffusion weighting factor, S(b) the DW signal with non-zero diffusion weighting gradient along certain direction, S(0) the signal without any diffusion weighting and D the diffusion coefficient. This linear relationship between the logarithmic signal decay and D is true for unrestricted diffusion. DKI is a model independent approach that describes the DW signal decay without imposing any biophysical modeling. For simplicity, consider the application of diffusion encoding gradients with infinitesimally short duration d and amplitude g Figure 1. Probability density function of a Gaussian (solid blue, K ¼ 0), normalized square of the Gaussian (dashed red, K > 0) and normalized square root of the Gaussian (dotted black, K < 0). NMR Biomed. 2010; 23: Copyright ß 2010 John Wiley & Sons, Ltd. View this article online at wileyonlinelibrary.com 837

3 E. X. WU AND M. M. CHEUNG along a particular direction in a PGSE experiment, the DW signal S is given by (12): SðgÞ Sð0Þ ¼ Z 1 1 e igdgr Pðr; DÞdr (3) where g is the gyromagnetic ratio, P(r,D) the displacement probability with net displacement r along that direction and diffusion time D. The logarithm of the signal attenuation can be expanded as a summation of the cumulants k p of P(r,D) (20,40): ln SðgÞ Sð0Þ ¼ X1 p¼1 k p ðiggdþ p p! When the diffusion is not symmetrical, the phase of the DW signal can be non-zero in theory (19,41,42). However, the voxel dimension in typical MR diffusion measurements is much larger than the diffusion distance and the scale of tissue microstructues. Thus the asymmetry of displacement probability profile can be neglected in practice (19,40). All odd order cumulants are therefore null and Eq. (4) can be expanded as: ln SðgÞ Sð0Þ ¼ k ðggdþ (4) ðggdþ 4 ðggdþ 6 þ k 4 þ k 6 þ ::: (5) 4! 6! Given that diffusion coefficient D for isotropic Gaussian diffusion in time D is defined by (12): D ¼ k 2 2D ; (6) its substitution into Eq. (1) gives: k 4 ¼ 4KD 2 D 2 : (7) Diffusion weighting factor b in PGSE sequence is defined as: b ¼ g 2 g 2 d 2 D: (8) Substituting Eqs. (6)-(8) into Eq. (5), the signal attenuation can be approximated by the quadratic exponential kurtosis model after truncating the 3rd term and above (20): ln SðbÞ Sð0Þ bd þ 1 6 b2 D 2 K: (9) By measuring DW signals with multiple b-values, we can estimate the apparent diffusivity (D app ) and apparent diffusion kurtosis (K app ) along a specific diffusion direction by fitting Eq. (9) (20). Diffusivity tensor (DT) derived parameters In an anisotropic 3D medium, distribution of D app can be formulated as a 2nd-order 3D diffusivity tensor and represented by a diffusivity ellipsoid as in conventional DTI (1,2). One can compute the rotationally invariant DTI indices such as l //, l?,md and FA from the three DT eigenvalues (l 1 > l 2 > l 3 ) as follows (3,43 48): l == ¼ l 1 ; (11) l? ¼ l 2 þ l 3 ; (12) 2 MD ¼ l 1 þ l 2 þ l 3 ; 3 (13) rffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ðl 1 MDÞ þðl 2 MDÞ þðl 3 MDÞ FA ¼ 2 l 2 1 þ l2 2 þ : l2 3 (14) Kurtosis tensor (KT) derived parameters The description of the 2nd kurtosis term in Eq. (9) requires the 4th-order 3D KT W ijkl.k app estimated along an arbitrary direction using Eq. (9) is related to W ijkl by (20): K app ¼ MD2 D 2 X3 X 3 X 3 X 3 n i n j n k n l W ijkl : (15) app i¼1 j¼1 k¼1 l¼1 where n i is the ith element of the diffusion direction. Due to the 3D symmetry of diffusion processes probed by MR, KT contains only 15 independent components and Eq. (15) can be reduced to (21,49): K app ¼ MD2 B 4! D 2 app 0 P 3 i¼1 n 4 i W iiii þ 4! 3! P 3 P 3 i¼1 j¼1 j6¼i n 3 i n jw iiij þ 2!2! 4! P 3 P 3 i¼1 j¼1 j6¼i W iijj þ 4! 2! P 3 P 3 P 3 W iijk i¼1 j¼1 k¼1 j6¼i k6¼i k6¼j ¼ MD2 D 2 n 4 1 W 1111 þ n 4 2 W 2222 þ n 4 3 W 3333 app þ 4n 3 1 n 2W 1112 þ n 3 1 n 2W 2223 þ n 1 n 3 2 W 2221 þ n 3 2 n 3W 2223 þ n 1 n 3 3 W 3331 þ n 2 n 3 3 W 3332 þ 6n 2 1 n2 2 W 1122 þ n 2 1 n2 3 W 1133 þ n 2 2 n2 3 W 2233 þ 12 n 2 1 n 2n 3 W 1123 þ n 1 n 2 2 n 3W 1223 þ n 1 n 2 n 2 3 W i C A (16) Note that when d is not negligible compared with D, the effective diffusion time should be expressed as (D-d/3) and Eq. (8) should be (12,39): b ¼ g 2 g 2 d 2 D d : (10) 3 As a result, K app estimates using Eq. (9) along at least 15 non-collinear and non-coplanar directions are required to construct KT (W ijkl ). Mean kurtosis (MK) has been introduced in the original DKI study (20). It is computed as the average kurtosis along all uniformly distributed diffusion directions: 838 View this article online at wileyonlinelibrary.com Copyright ß 2010 John Wiley & Sons, Ltd. NMR Biomed. 2010; 23:

4 DIFFUSION KURTOSIS IMAGING OF NEURAL TISSUE MK ¼ 1 n MK is a measure of the overall kurtosis. It does not have any directional specificity. To explore the 3D characteristics of the KT, Lu et al. (21) decomposed the kurtosis 3D distribution into spherical harmonics and computed the 0th-, 2nd- and 4th-order harmonics. The 0th-order harmonic is the component of isotropic 3D distribution and closely related to the MK above. Although such spherical harmonic analysis is useful to reduce the 3D angular plot into simple indices (21,50,51), these indices lack the direct physical relevance to the diffusion processes. Given the mathematical complexity of the 4th-order tensor (52), interpretation of individual KT elements, eigenvalues and eigenvectors are yet to be explored (22,53 55). Directional kurtosis analysis has been proposed recently to examine the diffusion kurtosis along a specific direction, including the direction parallel or perpendicular to the principal DT eigenvector, i.e., axial or radial direction (22,33,56). To compute such directional kurtoses, the KT is first transformed from the standard Cartesian coordinate system to another coordinate system formed by the three orthogonal eigenvectors of the DT (22,53). With a rotation x ¼ Pn, bw ijkl ¼ X3 X n i¼1 X 3 X 3 X 3 i 0 ¼1 j 0 ¼1 k 0 ¼1 l 0 ¼1 K app i : (17) e i 0 ie j 0 ie k 0 ie l 0 iw i 0 j 0 k 0 l 0 (18) where e ij are elements of the 3D rotation matrix P. The kurtosis along an individual DT eigenvector direction can then be computed from this KT transformation similar to Eq. (15), K i ¼ MD2 l 2 bw iiii (19) i A graphical illustration is shown in Figure 2 where the 3D diffusion distribution is represented by a diffusivity ellipsoid (blue) with eigenvalues l 1, l 2 and l 3 as the 3 axes, respectively. Note that the 3D kurtosis distribution, as characterized by KT as in Eq. (16), is complex and illustrated by a simple oblate ellipsoid. K 1,K 2 and K 3 are the kurtosis along eigenvector e 1,e 2 and e 3, respectively. Axial kurtosis (K // ) and radial kurtosis (K? ) can then be defined as the kurtosis parallel and perpendicular to the principle diffusion eigenvector (e 1 ), respectively, by (22): K == ¼ K 1 ; (20) K? ¼ K 2 þ K 3 : (21) 2 Note that, in contrast to Eq. (13), MK is not necessarily equal to (K // þ 2K? )/3 because 3D kurtosis distribution cannot be represented by a simple ellipsoid. Similar to FA in DTI, the anisotropy of directional kurtosis can be conveniently defined as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2þ 2þ 2 3 FA K ¼ 2 K 1 K K2 K K3 K K 2 1 þ K2 2 þ ; (22) K2 3 where K ¼ 1 P 3 3 K i. Here, FA K is derived from the three directional i¼1 kurtoses (K 1,K 2 and K 3 ) only. It is an arbitrary measure of the anisotropy of 3D kurtosis distribution. APPLICATIONS IN NEURAL TISSUE CHARACTERIZATION Conventional DTI quantitation vs b-values Figure 3a shows the typical normalized DW signal decay curves observed in two regions (cerebral peduncle and cerebral cortex) of an adult rat brain in vivo, exhibiting the effect of non-gaussian diffusion in both WM and GM. Non-monoexponential b-value dependency has been reported in biological tissues with the use of higher b-values (5,8 10,18,19,57,58). Both restricted diffusion and voxel heterogeneity can lead to the deviation of water diffusion profile from a Gaussian. Clearly, conventional DTI with a single non-zero b-value is not adequate to fully characterize the signal decay. The error bars in Figure 3a indicate the standard deviation (SD) along 30 diffusion gradient directions. Larger SD is found in WM than in GM because of the lack of preferential diffusion direction in GM. Figure 3b shows the conventional DTI-derived parametric maps using different b-values. As a result of the diffusion restriction, the directional and mean diffusivities generally decrease with b-value and FA quantitation is also influenced to various extents depending on the brain structures concerned. The contrasts among WM, GM and cerebrospinal fluid (CSF) in l //, l?, MD and FA maps change with b-value. As the extent of non-monoexponential decay is dictated by the complex underlying cellular microstructures, the conventional DTI quantitation varies with b-value used and extent of variations depends on both neural tissue microstructural characteristics and orientation (59). Consequently, the ability of conventional DTI in detecting microstructural alterations also changes with b-value (60). Therefore, the comparison among different DTI studies must be made with caution. DKI parametric maps Figure 2. The 3D diffusion distribution is represented by a tri-axial diffusivity ellipsoid (blue) with e 1, e 2 and e 3 as the eigenvectors of the 2 nd -order 3D diffusivity tensor (DT). Note that the 3D kurtosis distribution is complex and dictated by the 4 th -order 3D kurtosis tensor (KT) as described in Eq. (16). For simplicity, it is depicted as a tri-axial oblate ellipsoid (green) here. Figures 4a and 4b show the typical KT- and DT-derived parametric maps (K //,K?, MK, FA K, l //, l?, MD and FA) from in vivo and formalin-fixed adult rat brains, respectively (22). DKI parametric maps obtained from a normal human subject are also shown in Figure 4c. Higher MK is found in WM, indicating a generally higher degree of diffusion complexity and restriction in the WM structures. It can be seen from the directional kurtosis maps that such high MK in WM is mainly contributed by K?. This suggests NMR Biomed. 2010; 23: Copyright ß 2010 John Wiley & Sons, Ltd. View this article online at wileyonlinelibrary.com 839

5 E. X. WU AND M. M. CHEUNG the existence of heterogeneity and restricted diffusion in axonal structures (5,6,61,62). Both MK and K? exhibit strong contrast between WM and GM structures. Positive mean and directional kurtoses are observed in both WM and GM, indicating faster DW signal decay at lower b-values and restricted diffusion environment in both WM and GM under in vivo and formalin-fixed conditions. Alteration of cellular structures has been noted for histological fixation (63 65), and hence diffusion is restricted in fixed neural tissues to a different extent along axial and radial directions. Quantitative DKI analysis shows a general decrease in diffusivity Figure 3. a) Normalized DW signal decay curves observed in WM cerebral peduncle (CP) and GM cerebral cortex (CT) from a Sprague Dawley rat in vivo. Regions of interest (ROIs) are defined on a single coronal slice. S(b) is the average of all normalized DW signals along 30 uniformly distributed diffusion gradient directions. The error bar indicates the standard deviation (SD). b) Typical quantitative maps of FA, mean diffusivity (MD), axial diffusivity (l//) and radial diffusivity (l?) computed by conventional DTI using different 2-b-value sets (i.e., 0 versus 0.5, 1.0, 1.5, 2.0 or 2.5 ms/mm2) and all 6 b-values (0, 0.5, 1, 1.5, ms/mm2) via monoexponential fitting (Monoexp). Both parametric quantitation and contrast between WM and GM are seen to vary with the b-values used. The raw DWI data was acquired with a Bruker PharmaScan 7T scanner using 4-shot spin-echo (SE) echo planar imaging (EPI) with TR/TE ¼ 3000/ 30.3 ms, d/d ¼ 5/17 ms, slice thickness ¼ 1 mm, FOV ¼ mm2, data matrix ¼ (zero-filled to ) and NEX ¼ 4. " Figure 4. Typical DKI-derived parametric maps from a single slice of a) in vivo, b) formalin-fixed adult rat brains and c) a normal human subject (male, 44 years old). Axial diffusivity (l//), radial diffusivity (l?), mean diffusivity (MD), axial kurtosis (K//), radial kurtosis (K?), mean kurtosis (MK), fractional anisotropy (FA), directionally encoded colour FA (DEC-FA) and fractional anisotropy of kurtosis (FAK) maps are computed from DKI model. MK, K? and FAK maps show strong contrasts between WM and GM. These contrasts are seen to vary among in vivo and ex vivo conditions. All diffusivities are displayed in mm2/ms. For (a), raw DWIs were acquired by SE EPI with TR/TE ¼ 3000/30.3 ms, d/d ¼ 5/17 ms, slice thickness ¼ 1 mm, FOV ¼ mm2, data matrix ¼ (zero filled to ), NEX ¼ 4, 6 b-values (0.0, 0.5, 1.0, 1.5, 2.0 and 2.5 ms/mm2) and along 30 directions using 7T scanner (22). For (b), raw DWIs were acquired with the same parameters as those for in vivo except TE ¼ 34.3 ms, d ¼ 9 ms and b-values of 0.0, 1.0, 2.0, 3.0, 4.0 and 5.0 ms/ mm2. A larger b-value range was used in ex vivo experiment due to the generally lower diffusivities. For (c), raw DWIs were acquired by SE EPI with TR/TE ¼ 2300/109 ms, slice thickness ¼ 2 mm, FOV ¼ mm2, data matrix ¼ , NEX ¼ 2, 6 b-values (0.0, 0.5, 1.0, 1.5, 2.0 and 2.5 ms/ mm2) and along 30 directions using a 3T Siemens scanner (32). 840 View this article online at wileyonlinelibrary.com Copyright ß 2010 John Wiley & Sons, Ltd. NMR Biomed. 2010; 23:

6 DIFFUSION KURTOSIS IMAGING OF NEURAL TISSUE and increase in kurtosis in WM and GM of the fixed brains (22). The varying extent of diffusivity decrease in WM and GM, with FA values largely preserved, has been reported in previous DTI studies of formalin-fixed rodent and primate brains (63 65). The breakdowns of neurofilaments and microtubules caused by fixatives (66 70) are believed to produce more diffusion barriers and hence lead to the l // decrease and K // increase. Other fixation effects such as tissue shrinkage (69), decrease in membrane permeability (70), increase in axonal packing density (66) and reduction of extracellular space in parenchyma (71) also likely contribute to the significant l? decrease and K? increase. Although these microstructural alterations may account for the diffusivity and kurtosis differences observed between in vivo and formalin-fixed brains, the effect of formalin fixation cannot be evaluated fully in this particular study because different temperatures were used for in vivo and ex vivo experiments and diffusion is known to be affected by temperature (65,72,73). Directional kurtosis analysis of fixed experimental autoimmune encephalitis (EAE) spinal cord The inflammatory neurodegenerative disease EAE resembles multiple sclerosis (MS) in many aspects and is characterized by both axonal loss and demyelination. DTI experiments have been applied to differentiate the EAE or MS pathology along axial and radial directions and DTI indices have been demonstrated to be more sensitive than conventional MRI imaging (74 77). In recent DKI studies, there are promising results of using MK to detect changes in normal or pathological neural tissue (20,21,30 32,78). However, as an average of kurtoses along all the diffusion directions, MK can lose sensitivity and specificity in probing directional changes of pathological tissue. Figure 5 shows the DKI analysis of formalin-fixed EAE-induced rat spinal cord (SC) (56), demonstrating a better differentiation and direction specific description of the pathological changes by directional diffusivity and kurtosis analysis. Significant changes can be detected in directional diffusivities and kurtoses but not in MK. K // is found to be significantly increased and l // decreased in the lesion area. l // reduction is likely due to cytoskeletal perturbation or debris formation when the axonal structures break down (79). In addition, l? increases whereas K? decreases likely because of the demyelination and axonal loss that also lead to less diffusion restriction in radial direction. The directionally averaged MD and MK are found to be less sensitive to EAE pathology due to the opposite trends of diffusivity and kurtosis changes in axial and radial direction. Monitoring postnatal brain maturation by DKI and conventional DTI DTI has been applied to assess the subtle morphological changes in various human (80 87) and rodent (88 95) developmental brain studies. Postnatal normal brain development is known to be temporally accompanied with gradual and local tissue morphological changes in both WM and GM that can lead to subtle changes in water diffusion (80,83 85,90,93 97), thus providing an effective biological platform to evaluate the sensitivity of DKI. An in vivo DKI study has been recently performed to examine the changes in various diffusivity and kurtosis parameters at different postnatal time points (33,60). DWIs acquired in this study have been analyzed using both conventional DTI and DKI scheme. Figure 5. a) Typical DKI-derived parametric maps of a formalin-fixed EAE-induced rat spinal cord (SC). The red arrows indicate the WM lesions and blue arrows the normal WM. b) Pixel-based scatter plot of the directional diffusivities and kurtoses measured within the normal and lesion ROIs in a single slice from each EAE SC sample (n ¼ 4). ROIs have been defined on T2W images. Lesions can be clearly differentiated especially by direction kurtosis analysis. c) ROI analysis of DT- and KT-derived parameters in the normal WM and lesions. Percentage differences are shown with p < 0.05 indicating statistically significant by Mann Whitney s test between normal and lesion tissues. All image data were acquired by SE 8-shot EPI with TR/TE ¼ 3000/45 ms, d/d ¼ 9/17 ms, slice thickness ¼ 2 mm, FOV ¼ mm 2, data matrix ¼ (zerofilled to ), NEX ¼ 2, 5 bvalues (1.2, 2.4, 3.6, 4.8, 6 ms/mm 2 ) and along 30 directions using a 7T scanner (48). Figure 6 shows how the sensitivity of conventional DTI changes with the b-value used (0.5, 1.5 and 2.5 ms/mm 2 ). The number of statistically significant differences detected for the 4 WM and 3 GM structures among the 3 time points are indicated in upper right corner of each plot. It can be clearly seen that the choice of NMR Biomed. 2010; 23: Copyright ß 2010 John Wiley & Sons, Ltd. View this article online at wileyonlinelibrary.com 841

7 E. X. WU AND M. M. CHEUNG Figure 6. In vivo l //, l?, MD and FA measured in different rat brain WM and GM structures among 3 age groups (n ¼ 6 each), postnatal day 13 (P13), 31 (P31) and 120 (P120). They are computed by conventional DTI scheme using different b-values (in ms/mm 2 ). Error bars represent the interanimal SDs in each age group. The statistical comparisons between 3 age groups are performed with Tukey s test after one-way ANOVA. The total number of significant changes detected by each parameter is indicated in the upper right corner of each plot. The original DWIs were acquired by 4- shot SE-EPI sequence with different b-values (0.0, 0.5, 1.0, 1.5, 2.0, 2.5 ms/mm 2 ) along 30 directions. The imaging parameters were similar to those for Fig. 4a and adjusted slightly for resolution and SNR for P13 rats (34). CC: corpus callosum; EC: external capsule; CP: cerebral peduncle; AC: anterior commissure; CT: cerebral cortex; HP: hippocampus; CPu: caudate putamen. 842 b-values affects the ability of DTI indices to monitor the tissue changes to varying extent. The sensitivity of l // in detecting rat brain WM maturation is generally observed to be the highest at low b-value as assessed by the total number of statistical significances whereas that of l? is the highest at high b-value. At relatively low b-values, the apparent diffusivity is primarily contributed from the fast water diffusion activities in extracellular space that depend on both cellular microstructure and membrane permeability (5,9,98,99). The use of low b-value can best detect these changes. The high l // sensitivity at low b-value observed in the current study suggests the alterations of these fast water diffusion activities along axonal direction during brain maturation. Such alterations may result from the increase in packing density of fiber bundles and axons, axonal diameter increase, changes in neurofibrils, and increased complexity of extracellular matrix (83,85). On the other hand, the diffusion changes probed in WM using high b-values are ascribed more to the slow water molecule diffusion particularly along the radial direction when traversing the membranes and myelin sheaths (22,99). The high sensitivity of l? at high b-value in detecting brain maturation shown in Figure 6 likely reflects these WM microstructural changes, including myelination and axonal density and diameter changes during postnatal brain development. FA quantitation is also affected by the b-value and its ability in detecting brain maturational changes varies among different structures. Note that several previous studies have reported such b-value dependency of DTI in quantifying neural tissues (59,60). Figure 7 shows the corresponding ROI measurements of various diffusivity and kurtosis parameters that are derived from all DWIs using the DKI model. The same set of DWIs is also fitted into the monoexponential DTI model for comparison. Figure 7a shows that the sensitivity of fitting all the multi-b-value DWIs to DTI model is generally similar to that of employing a medium b-value (b ¼ 1.5 ms/mm 2 ) shown in Figure 6. In Figure 7b, the general and continual kurtosis increase with age is observed, indicating that more diffusion restriction occurs during brain maturation in both WM and GM structures. The DKI-derived diffusivity and kurtosis indices are highly sensitive to brain developmental changes. Both l // and K // of WM are found to increases with age, which may arise from various biological events during early postnatal brain maturation. The increase of diffusivity can be caused by axoplasmic flow during the myelination period (81). At the same time, neuronal loss and axonal pruning that shortens the axon length can lead to an increase of restriction (95,100). The increase of K? in WM is likely ascribed to the myelination and modification of axonal structures that increases restriction in the radial direction. Although MK is sensitive in detecting rat brain maturation, directional kurtosis here provides more directionally specific information. DKI analysis also reveals that diffusion restriction in the relatively isotropic GM increases with age. This may reflect the more densely packed structures and the dendritic architectural modification in GM (85). Note that the directional diffusivities estimated by the monoexponential DTI model (in Fig. 7a) reflect the combined and sometimes competing effect of diffusivity and kurtosis. When there is a large K, the estimated diffusivity in conventional DTI shows a large discrepancy with the diffusivity estimated in DKI approach. As K in all the structures is positive, DTI-derived diffusivities are generally lower than those by DKI. The relatively high sensitivity of the l? in monoexponential DTI model is mainly a result of increasing K? with age (while the changes of l? in DKI View this article online at wileyonlinelibrary.com Copyright ß 2010 John Wiley & Sons, Ltd. NMR Biomed. 2010; 23:

8 DIFFUSION KURTOSIS IMAGING OF NEURAL TISSUE Figure 7. Corresponding DKI-derived parameters measured in different rat brain WM and GM structures in 3 age groups. Multiple-b-value and multiple-direction DW signals were first fitted to Eq. (9) for D app and K app along each direction. Diffusion and kurtosis tensors were then computed and various parameters calculated. The imaging parameters were the same as those described in Fig. 6 caption. are moderate). DTI-derived l // is related to the increase of both K // and l // derived in DKI that manifests opposite and competing effects. Therefore diminished sensitivity in detecting maturational changes of l // in conventional DTI are observed. The separation of l // and K // can improve the characterization of neural tissue along the axial direction. Because the complex biological modification of WM along axonal direction affects both diffusivity and kurtosis, information obtained in conventional DTI is inadequate to fully infer the microstructural changes during brain maturation. Other applications As a more comprehensive model to describe the restricted diffusion process in vivo, DKI is potentially a valuable tool in probing pathological alterations in neural tissues. Our recent studies have demonstrated that DKI, together with directional analysis, could lead to improved neural characterization with better sensitivity and directional specificity. One fundamental limitation of conventional DTI is lack of sensitivity in detecting GM changes, and it is worth investigating further whether kurtosis can better probe subtle diffusion changes in GM. It has been reported recently that training-related changes in WM structure can be revealed by DTI (101,102). In order to study density changes in GM, voxel-based morphometry (VBM) is commonly employed. DKI may serve as a more sensitive tool to detect and characterize such subtle changes in both WM and GM. Recent preliminary studies have reported the ability of DKI to monitor neural tissue changes as a function of age (32). Distinct MK patterns were found to be age-related and MK could be used to NMR Biomed. 2010; 23: Copyright ß 2010 John Wiley & Sons, Ltd. View this article online at wileyonlinelibrary.com 843

9 E. X. WU AND M. M. CHEUNG 844 characterize both WM and GM changes. DKI has also been applied in various pathological states, including Alzheimer s disease (29), schizophrenia (30) and attention deficit and hyperactivity disorder (31). In these studies, kurtosis has been shown to change with pathological alterations. More recent and preliminary studies have also reported the ability of directional analysis of kurtosis in detecting specific microstuctural changes in chronic mild stress (36) and Huntington s disease (35). DKI has also been sought to resolve the crossing of WM fibers (103) and possibly lead to more accurate tracking and characterization. Given that cellular WM and/or GM microstructures are altered in many diseases, DKI is potentially applicable to a variety of neurological disorders such as epilepsy, ischemic stroke, both WM and GM abnormalities in MS, Parkinson s disease, Alzheimer s disease and other forms of degenerative dementia, and brain tumors under both preclinical and clinical settings. It should be noted that the number of such DKI investigations is still limited, more comprehensive studies are desired to further validate and explore the merits of DKI approach in characterizing various neural tissue alterations. DKI QUANTITATION ISSUES DKI quantitation DKI can be readily implemented on most clinical scanners. DWIs with multiple b-values along at least 15 non-collinear and non-coplanar diffusion directions are required in DKI. In the aforementioned studies, at least five different b-values have been used. In fact, a minimum of three b-values (including b ¼ 0) are sufficient to fit Eq. (9) and a more efficient and clinically feasible acquisition scheme has been proposed to shorten the scan time (104). To obtain reliable fitting for D app and K app, a sufficient b-value range must be chosen to permit as much nonmonoexponential decay as possible. However, it must be before the occurrence of the minimum in Eq. (9), after which the kurtosis model will fail because bd app þ 1 6 b2 D 2 app K app will increase with b-value instead (22,105). In other words, b-values should be smaller than b min imum ¼ 3=ðD app K app Þ (22). Clinically, b-values no higher than 2.5 ms/mm 2 is sufficient for performing in vivo brain DKI (20,103). Note that b-values have been arbitrarily spaced and 30 uniformly distributed diffusion directions have been employed in all DKI studies so far. A recent study has shown that the distribution of diffusion directions and b-values in DKI can be optimized systematically to increase the precision of the kurtosis estimation though more experimental validations are yet to be performed (106,107). Another key factor in DKI quantitation is the diffusion time (D) used in acquiring DWIs. Diffusion time is known to be an important factor in characterizing diffusion. The water displacement profile has been studied by q-space imaging to investigate its changes with D (58, ). In restricted diffusion, there is a turning point of the root-mean-square displacement at a particular D. It has been proposed that this turning point can be used to determine the size of the neural fibers if the diffusion direction is perpendicular to the long axis of the microstructure ( ). At large D, the slow diffusing component primarily originates from the intra-axonal space where axonal membranes and myelin act as barriers, leading to changes in apparent water diffusion displacement profile. Therefore, the kurtosis quantitation as determined by DKI model is expected to vary with D as well. Figure 8a shows the typical normalized DW signal decays as a function of various diffusion times in 2 WM and 2 GM structures from a rat brain in vivo. In all structures, it can be seen that the decay curves in WM tilt upward as D increases, leading to the general MD decrease with D as shown in Figure 8b. MK also depends on D. Note that these kurtosis changes are related to both the diffusion restriction dimension and water exchange between compartments (5,61,114). These two factors manifest competing effects on how DW signal varies with D (114). The MK decrease and plateau at long D may provide information in inferring microstructural properties of neural tissues. Nevertheless, these experimental results clearly demonstrate that kurtosis quantitation can be diffusion time dependent. In other words, diffusion time influences the actual effect of diffusion restriction vs compartmental water exchange on DKI measurements. Interpretation of DKI-derived parameters In DKI, both diffusivity and kurtosis parameters are estimated. Because of the generally positive kurtosis observed in tissues, the mean and directional diffusivities determined by DKI are generally lower than those reported using conventional DTI (with b ¼ 1.0 ms/mm 2 ) (22). The combined and multi-parametric analysis of various diffusivity and kurtosis indices will undoubtedly offer improved neural tissue characterization over the conventional DTI analysis though the optimal analysis strategy and validation are yet to be investigated in the future. As in DTI, DKI-derived quantities can be affected by signal to noise ratio (SNR) in DWIs. The effect of noise on DTI-derived parameters has been studied ( ). In DKI, the quantitative effect of SNR remains to be studied. Nevertheless, SNR must be maintained sufficiently high in DKI for reproducible interpretation given the higher b-values used, multi-parameter curve fitting of Eq. (9) and estimation of 15 KT components from Eq. (16). In theory, at least 15 independent diffusion encoding directions are required to determine KT in contrast to 6 as required for DT. Thus one would expect that intrinsically noise will affect the KT-derived parameters more than the DT-derived ones. In practice, CSF partial volume effect can lead to false diffusion kurtosis in voxels containing both tissue and CSF. In presence of CSF contamination, more than one diffusion compartments will exist and kurtosis will be overestimated and misinterpreted as an increased diffusion restriction or tissue complexity. Thus care must be taken when interpreting kurtosis data from voxels with CSF component. The effect of CSF partial volume in diffusion quantitation using Eq. (2) and Eq. (9) remains to be investigated. It has been reported that MK is less vulnerable to CSF contamination when compared with other DTI parameters such as MD and FA derived in DKI model (118). This is likely due to the fact that the early rapid DW signal decay described by term -bd app in Eq. (9) will be affected more, leading to more variations with respect to CSF partial volume effect. Finally, DKI is a model-independent approach that does not assume any biophysical property like water exchange (114,119) or compartmentalization (8,9). Although it is difficult to relate the kurtosis to a specific biophysical property, DKI model reduces the complex in vivo diffusion process into the first and second terms in Eq. (9) for Gaussian and non-gaussian component, respectively. Furthermore, Eq. (9) is a truncated logarithmatic expansion of DW signal decay that only includes the 2nd- and 4th-order cumulants. Higher order terms are omitted based on the assumption that View this article online at wileyonlinelibrary.com Copyright ß 2010 John Wiley & Sons, Ltd. NMR Biomed. 2010; 23:

10 DIFFUSION KURTOSIS IMAGING OF NEURAL TISSUE Figure 8. DKI measurements of mean diffusivity (MD) and kurtosis (MK) vs. diffusion time D used. a) Normalized DW signal decay curves of 2 WM and 2 GM structures computed by averaging DW signals along all 30 diffusion gradient directions from an adult rat brain in vivo. b) ROI measurement of MD (in mm 2 /ms) and MK with error bars indicating the SD among four adult Sprague Dawley rats studied (n ¼ 4). DWIs were acquired by 2-shot stimulated echo EPI with 5 different b-values (0.0, 0.5, 1.25, 2, 2.5 ms/mm 2 ) along 30 gradient encoding directions. The imaging parameters were TR/TE ¼ 3000/25.1 ms, d ¼ 6 ms, slice thickness ¼ 1.2 mm, FOV ¼ mm 2, data matrix ¼ (zero-filled to ) and NEX ¼ 2 at 7T. Six diffusion times were studied (40, 70, 95, 125, 150 and 190 ms). CC: corpus callosum; IC: internal capsule; CT: cerebral cortex; CU: caudate putamen. they are insignificant. Although such truncation prevents us from fully describing the deviation of diffusion displacement profile from Gaussian distribution, it leads to the estimation of kurtosis of diffusion profile. As kurtosis is the 1 st -order description of the deviation from Gaussian or free diffusion, it can serve as a valuable and yet relatively simple marker for the overall non-gaussian and restricted diffusion in complex biological tissues in clinical settings. CONCLUSION In summary, diffusion kurtosis imaging provides a higher-order description of the water diffusion process in vivo by a 2nd-order 3D diffusivity tensor as in conventional DTI together with a4 th -order 3D kurtosis tensor. DKI relates DW signal decay to the b-value by both the diffusivity and the kurtosis in a quadratic exponential manner. Because kurtosis is a measure of the NMR Biomed. 2010; 23: Copyright ß 2010 John Wiley & Sons, Ltd. View this article online at wileyonlinelibrary.com 845

11 E. X. WU AND M. M. CHEUNG deviation of the diffusion displacement profile from a Gaussian distribution, DKI analyses quantify the degree of diffusion restriction or tissue complexity. Recent studies have demonstrated that DKI offers a more comprehensive approach than DTI in describing the complex water diffusion process in vivo. By quantifying both mean and directional kurtoses and diffusivities, DKI may provide improved sensitivity and specificity in MR diffusion characterization of neural tissues. Acknowledgements This work was supported by the Hong Kong Research Grant Council (RGC GRF HKU7808/09M). We thank Dr Edward S. Hui, Mr Kevin C. Chan and Dr Wutian Wu of University of Hong Kong, and D Liqun Qi of Hong Kong Polytechnic University for their technical assistance. We also thank Drs Joseph A. Helpern and Jens H. 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Does diffusion kurtosis imaging lead to better neural tissue characterization? A rodent brain maturation study. Neuroimage 2009; 45(2): Lätt J, Westen Dv, Nilsson M, Wirestam R, Ståhlberg F, Holtås S, Brockstedt S. Diffusion time dependent kurtosis maps visualize ischemic lesions in stroke patients. Proceedings of the 17th Annual Meeting of ISMRM, Honolulu, USA, p Blockx I, Verhoye M, De Groof G, Van Audekerke J, Raber K, Poot D, Sijbers J, von Horsten S, Van der Linden A. Diffusion Kurtosis Imaging (DKI) reveals an early phenotype (P30) in a transgenic rat model for Huntington s disease. Proceedings of the 17th Annual Meeting of ISMRM, Honolulu, USA, p Delgado Y, Palacios R, Verhoye M, Van Audekerke J, Poot D, Sijbers J, Wiborg O, Van der Linden A. DKI visualizes hippocampal alterations in the chronic mild stress rat model. Proceedings of the 17th Annual Meeting of ISMRM, Honolulu, USA, p View this article online at wileyonlinelibrary.com Copyright ß 2010 John Wiley & Sons, Ltd. NMR Biomed. 2010; 23: